Abstracts

Axiomatische Wahrheitstheorien, Akademie Verlag, Berlin, 1996

In this book I provide a technical survey of many axiomatic theories of truth as well as a survey of many semantical approaches to the paradoxes.

I Grundlagen
1 Explikationen von Wahrheit
2 Das Kategorieproblem
3 Metamathematik der Wahrheit
II Die Basistheorie
4 Rekursive Funktionen
5 Arithmetisierung
6 Beweistheoretische Reduktionen
III Klassische Wahrheitstheorien
7 Partielle Wahrheitsprädikate
8 Wahrheitstheorien für PA
9 Konservativität
10 Wahrheit und arithmetische Komprehension
IV Wahrheitsklassen
11 Nonstandardmodelle
12 Lachlans Theorem
13 Partielle Erfüllungsklassen
14 Vollständige Wahrheitsklassen
V Iterierte Wahrheitstheorien
15 Objekt- und Metasprachen
16 Tarski-Hierarchien
17 Definierbarkeit in Tarski-Hierarchien
18 Ein System iterierter Wahrheit
19 Unfundierte Hierarchien
20 Burges Hierarchien
VI Typfreie Wahrheit
21 Selbstreferentielle Wahrheit
22 Einige Inkonsistenzen
23 Typfreie Tarski-Bikonditionale
VII Kripkes Wahrheitstheorie und ihre Axiomatisierungen
24 Kripkes Wahrheitstheorie und Tarski-Hierarchien
25 Kripke-Feferman-Theorien
26 Supervaluation
VIII Systeme vollständiger und konsistenter Wahrheit
27 Klassische symmetrische Wahrheit
28 Revisionssemantik und FS
29 Beweistheorie von FS
IX Philosophische Aspekte
30 Kompositionalität
31 Der reflektive Abschluß von PA
33 Ontologische Reduktion

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Semantics and Deflationism, unpublished habilitation thesis, 2001

In this book a develop a deflationist approach to semantics where truth and reference are treated as logico-mathematical notions.

1 Introduction
1.1 The project
1.2 Definitions and Axiomatizations
1.3 To what category do objects belong which can be true?
1.4 A Sketch of Modalized Disquotationalism
2 The History of Disquotationalism and Deflationism
2.1 Frege
2.2 Ramsey
2.3 Tarski
2.4 Strawson
2.5 Ayer
2.6 Quine
2.7 Williams
2.8 The Prosentential Theory
2.9 Horwich
2.10 Field
2.11 Summary
3 Material Disquotationalism
3.1 The Status of the Disquotation Sentences
3.2 Proving Infinite Conjunctions
3.3 Expressing Infinite Conjunctions
3.4 The Adequacy of Material Disquotationalism
3.5 The Failure of Material Disquotationalism
4 Modalized Disquotationalism
4.1 Truth Analyticity
4.2 Reference and Satisfaction
4.3 The Challenge by Substantial Theories
5 Refinements
5.1 Truth in Foreign Languages?
5.2 Shapiro’s Puzzle
5.3 Indexicals, Demonstratives and Time
6 Lewys Argument
6.1 The History of Lewy’s Argument
6.2 Consequences of Lewy’s Argument
6.3 The Propositionalist Response
6.4 Modalized Disquotationalism and Propositionalism
7 Deflationism and Conservativeness
7.1 Conservativeness: Terminology and Basic Considerations
7.2 The Commitments of Deflationism
7.3 The Innocence of Deflationist Truth
8 Technical Appendix
8.1 Notation and Technical Preliminaries
8.2 The Disquotation Sentences
8.3 Satisfaction Classes
8.4 T-Analyticity
8.5 Typed Disquotation
8.6 Untyped Disquotation
8.7 Interpretation of AT+ in KF
8.8 Interpretation of KF in AT+
8.9 Some Corollaries
8.10 Open Questions: Conservativeness

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A system of complete and consistent truth, Notre Dame Journal of Formal Logic 35 (1994), pp. 311-327

The theory FS is formulated in the language of arithmetic plus a unary predicate symbol T for truth. FS is given by the axioms of Peano arithmetic plus all induction axiom in the language FS. The special axioms and rules for T are the following:

• axioms saying that truth commutes with all connectives and quantifiers. For instance, one axiom says that -A is true iff A is not true, where A is any sentence possibly containing T.
• From A one can conclude T[A], and vice versa. A is here again any sentence.

FS is shown to be equivalent to a system studied by Friedman and Sheard (1987). I follows from a result due to McGee (1985) that the system is omega-inconsistent.

FS may be seen as an axiomatization of the finite levels of rule-of-revision semantics. This insight is used in order to show that FS is proof-theoretically equivalent to ramified analysis for all finite levels. This implies that FS is arithmetically sound. If induction is restricted to arithmetical formulas, the resutling system is proved to be conservative over PA. The theory of satisfaction classes is employed in this argument (remark: a proof-theoretic argument for the conservativeness was given later in Halbach 1999).

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Tarski-hierachies, Erkenntnis 43 (1996), pp. 339-367

The hierarchy of truth predicates for all levels smaller than thefirst non-recursive ordinal is equivalent to the ramified analytical hierarchy up to the same level.

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Tarskian and Kripkean truth, Journal of Philosophical Logic 26 (1997), pp.69-80

An analysis of Kripke’s theory of truth with the Strong Kleene evaluation scheme at the least fixed is provided in terms of a Tarskian hierarchy of languages. This hierarchy has been studied in Halbach (1996). In particular, some results on definability in Kripke’s theory are obtained via a translation of the language with an untyped truth predicate into a language with indexed truth predicates.

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Conservative theories of classical truth, Studia Logica 62 (1999), pp.353-370

Some axiomatic theories of truth and related subsystems of second-order arithmetic are surveyed and shown to be conservative over their respective base theory. The truth theories are formulated in the language of PA expanded by an additional unary predicate symbol for truth. PA + “there is a satisfaction class” is the theory with the clauses of Tarski’s inductive definition of truth turned into axioms with only arithmetical induction axioms. The conservativeness over PA of PA + “there is a satisfaction class” and of the theory FS with arithmetical induction only of Halbach (1994) is established by purely finitisitc means.

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Disquotationalism and Infinite Conjunctions, Mind 108 (1999), pp.1-22

According to the disquotationalist theory of truth, the Tarskian equivalences, conceived as axioms, yield all there is to say about truth. Several authors have claimed that the expression of infinite conjunctions and disjunctions is the only purpose of the disquotationalist truth predicate. The way in which infinite conjunctions can be expressed by an axiomatized truth predicate is explored and it is considered whether the disquotationalist truth predicate is adequate f2 August, 2014a>

On Lehrer’s Principle of Trustworthiness, Erkenntnis 50 (1999), pp.259-272

According to the usual foundationalist picture of knowledge, basic beliefs provide the empirical input on which the whole empirical knowledge of a person should rely. There are convincing arguments showing that this picture is not completely convincing, because the proposed basic beliefs were not sufficient for providing foundations of knowledge or they were in need of further justification and therefore not really basic.

Now coherentism claims that the distinction of basic and non-basic beliefs is not sensible at all. There may be basic beliefs and coherentism does not have to deny this, but these possibly existing basic beliefs do not figure prominently in a sound account of epistemic justification. The rejection of basic beliefs as foundations of knowledge, however, seems to deprive of the empirical input. But if knowledge does not rest on basic beliefs providing the empirical input, how can external factors influence our knowledge at all? Coherentism seems prone to a conception where epistemic agents are completely isolated from the world.

Coherentists have developed several strategies to evade the isolation objection. Keith Lehrer’s approach to solve the problem became especially important and was thus discussed intensively.

In order to give a rough idea how Lehrer accomplishes a plausible picture of how we acquire empirical information, I will present an example. Suppose that I accept that there is a red rose. This assumptions implies in particular that I do not believe that there is a red rose because of wishful thinking etc; rather I came to believe it with the objective of accepting it just in the case it is true; but so far my belief lacks justification and is thus not yet knowledge. I cannot conclude (and thereby justify) from my acceptance alone that there is a red rose indeed. Given some additional information, however, I can and do usually infer this. The additional premises required may include that I am not dreaming, that normal daylight is present, and that I know roses. The common feature of all these premises is that I must be a reliable epistemic agent in the present kind of situation. Exactly this is expressed by the principle of trustworthiness proposed by Lehrer in his “Theory of Knowledge” (1990):

T . Whatever I accept with the objective of accepting something just in case it is true, I accept in a trustworthy manner.

He explains the benefits of this principle in the following way:

The consequence of adding principle T to my acceptance system is that whatever I accept is more reasonable for me to accept than its denial. It has the effect of permitting me to detach the content of what I accept from my acceptance of the content. My acceptance system tells me that I accept that p, accept that q, and so forth. Suppose I wish to justify accepting that p on the basis of my acceptance system telling me that I accept that p. How am I to detach the conclusion that p from my acceptance system? The information that I accept that p, which is included in my acceptance system, does not justify detaching p from my acceptance of it in order to obtain truth and avoid error. I need the additional information that my accepting that p is a trustworthy guide to these ends. Principle T supplies that information and, therefore, functions as a principle of detachment. It is the rule that enables me to detach the conclusion that p from my acceptance of p.

Although this account seems to be plausible at first, I will show that the principle of trustworthiness cannot be used as intended by Lehrer. By reformulating the principle of trustworthiness and by an appeal to Löb’s theorem I will show that only trivial instances of the principle are consistent with our basic assumptions on beliefs.

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Two proof-theoretic remarks on EA+ECT (with Leon Horsten), Mathematical Logic Quarterly 46 (2000), pp. 461-466

In this note two propositions about the epistemic formalization of Church’s Thesis (ECT) are proved. First it is shown that all arithmetical sentences deducible in Shapiro’s system of Epistemic Arithmetic (EA) from ECT are derivable from Peano arithmetic PA + uniform reflection for PA. Second it is shown that the system EA+ECT has the epistemic disjunction property and the epistemic numerical existence property for arithmetical formulas.

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Truth and Reduction, Erkenntnis 53 (2000), pp. 97-126

In this paper ontological commitments are attributed to axiomatic systems rather than interpreted languages, when ontological reductions in the abstract sciences are to be carried out. In consequence, persons make ontological commitments to abstract objects by accepting formal theories. This feature allows to investigate ontological commitment and reduction in a naturalized setting.

I consider subsystems of analysis, i.e., subsystems of second-order arithmetic extending Peano arithmetic. The ontological commitments of these systems to sets depend on the axioms postulating the existence of second-order objects.

Proof-theoretic investigations have shown that some of these subsystems are reducible to certain axiomatic theories of truth formulated in the language of Peano arithmetic augmented by a one-place predicate symbol for truth. Hence certain set existence principles may be superseded by assumptions on the truth predicate. These reductions are considered to be ontological reductions: ontological commitments to sets of natural numbers are replaced by commitments to semantical principles. On the other hand, inverse results are known, where commitments to semantical principles are replaced by ontological commitments. Therefore certain semantical principles are related to certain set existence assumptions. In particular, I relate the commitment to parameterless arithmetical, arithmetical, predicative and inductively definable sets to the semantical principles of the Tarskian equivalences, the recursive definition of truth, compositional semantics and Kripke’s semantics, respectively.

Disquotationalism Fortified,  in Circularity, Definitions, and Truth, Anil Gupta und Andre Chapuis (eds.), Indian Council of Philosophical Research, New Delhi, 2000, pp. 155-176

A variant of modalized disquotationalism is discussed. A sentence is said to be analytic in the truth predicate if and only if it is a logical consequence of the uniform T-sentences. It is claimed that the disquotationalist is committed to the view that T-analyticity parsed this way is sound, that is, if A is analytic in the truth predicate, then A. The resulting theory of truth turns out to be equivalent to the “Tarskian” theory of truth and is therefore much stronger than the theory of the pure T-sentences often associated with disquotationalism. Moreover, truth in foreign languages is discussed from a disquotationalist point of view.

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How Innocent is Deflationism?, Synthese 126 (2001), pp. 167-194

The deflationist pretends to advocate a simple, clear and thin concept of truth by explaining how the phrase is true’’ axiomatized in a certain way can serve certain purposes. Unlike the representative of substantial accounts of truth, he does not commit himself to any deep and worrisome doctrines about the world, facts, coherence or correspondence. Maybe the deflationist makes a hash out of the notion of truth, but at least, it seems, he does so without any deep and dark mumbo-jumbo. But is he as innocent as he looks?

Recently, several authors among them Horsten 1995, Shapiro 1998 and Ketland 1999 have reproached the deflationist with an alleged weighty consequence of the deflationist’s account. They have suggested that the deflationist has tied himself down to the conservativeness of his axioms and rules for truth.

At first sight this reproach seems plausible because deflationism is often characterized as a position opposed to substantial’ theories of truth, such as the correspondence theory. An insubstantial theory ought to have only insubstantial consequences; that is, the theory should not imply any new non-semantical consequences. In other words, the theory should not prove any new consequences not involving truth; technically, it should be conservative.

If the truth theory is conservative, it does not contribute any substantial insights to what we already know. Also, a proof of conservativeness yields a relative consistency proof. If the theory in which we reason is consistent and allows for the proof of conservativeness of the truth theory, then the truth theory is also consistent. Thus conservativeness is a desirable feature of a truth theory---especially in the eyes of a deflationist who advances a thin concept of truth.

Some deflationists may have given the impression that their truth theories yield only trivial consequences. However, I am not aware of any deflationist who has explicitly bound him- or herself to the doctrine that his theory of truth is conservative, before the opponents of deflationism broached the topic. Therefore, the latter were very cautious in their attributions or tried to provide arguments for their charge. Recently, however, one arch-deflationist, Field 1999 has come close to confessing a commitment to conservativeness by saying:

...there is no need to disagree with Shapiro when he says conservativeness is essential to deflationism’’

In this paper, I investigate whether the deflationist can agree with Shapiro without being in conflict with some theorems on formal truth theories.

Some theories of truth that the deflationists have advanced are not conservative indeed. That is, they do prove new non-semantical sentences and therefore yield, metaphorically speaking, new insights in non-semantical facts. Different formal arguments will be presented that show how the conservativeness requirement is violated by allegedly deflationist truth theories. Furthermore I shall evaluate these formal results with respect to the deflationist position. As will be shown, an exact formulation of the conservativeness requirement is very important.

In order to judge the objections against deflationism one has to clarify two intricate and controversial points: what set of axioms and rules is adequate as a theory of truth from a deflationist point of view? And what kind of conservativeness is at issue?

In this paper, two axiomatizations will mainly be considered: the traditional disquotation or T-sentences and the inductive defining clauses (a la Tarski) turned into axioms. Both axiom sets have been used by deflationists. However, it seems that many deflationists consider the pure T-sentences insufficient and prefer the inductive clauses. Whether the latter or the T-sentences are used makes a difference, since they behave in different ways with respect to conservativeness.

But what kind of conservativeness is at issue at all? Even if the deflationist is committed to conservativeness, it is unclear what kind of conservativeness he is committed. Some opponents of deflationism think that deflationist axioms and rules must not prove any (logically) contingent sentence not containing the truth predicate. That is, the deflationist’s truth system is required to be conservative over logic, while others think that he has pledged himself only to the conservativeness over a basic theory of expressions.

Clearly, conservativeness over logic guarantees the complete neutrality of the truth theory with respect to any non-semantical questions, while conservativeness over a theory of expressions (or their Gödel codes) renders the axioms neutral only with respect to questions that are left undecided by the theory of expressions. Therefore, conservativeness over logic is more desirable but also more likely to be out of the deflationist’s reach. Thus, opponents have to be careful about what kind of conservativeness they ascribe to the deflationist. Both kinds of conservativeness will be discussed in the following.

The plan of the paper is as follows. In the next somewhat technical section, I will explain the basic terminology and some basic results concerning conservativeness and make the above remarks more precise. In order to determine whether any kind of conservativeness follows from the deflationist account of truth, I will present the core doctrines of deflationism. I will check what particular axioms are appropriate as deflationist axioms for truth. Furthermore, it will be investigated whether anything about conservativeness follows from typical deflationist claims such as that truth is only a device of generalization or that truth is not a property. In two technical intermezzi I will recall some formal results relevant to the philosophical discussion on conservativeness. In the final sections the results of the findings of section 3 are surveyed.

It will turn out that deflationism is wrong if it implies that an adequate axiomatization of truth is conservative over logic. In order to establish this claim, I do not invoke the relatively complicated truth theories at issue in Shapiro 1998; rather conservativeness over logic already fails in the case of the theory of T-sentences for trivial reasons. The case of conservativeness over other theories is somewhat more involved. There is also little chance for the deflationist to attain this kind of conservativeness, as I will argue.

Disquotational Truth and Analyticity, Journal of Symbolic Logic 66 (2001), pp. 1959-1973

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences, yet not in the scope of negation, the system with the reflection schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF, and thus of ramified analysis up to epsilon_0.

Modalized Disquotationalism, in Principles of Truth, Volker Halbach & Leon Horsten, Dr. Hänsel-Hohenhaus, Frankfurt a.M., 2002, pp. 75-101

An axiom to the effect that all instances of the disquotation scheme are necessary is combined with axioms for necessity. The resulting theory overcomes deficiencies of traditional disquotationalist theories of truth. Finally the exact proof-theoretic strength of the theory and some of its variants is determined.

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War Descartes erkenntnistheoretischer Voluntarist?, Zeitschrift für philosophische Forschung 56 (2002), pp. 545-562

According to certain authors like Alvin Goldman and William Alston, normative epistemology presupposes epistemological voluntarism. This is the doctrine that epistemic behaviour like believing, judging, suspension of judgement and so on are voluntary actions. On this account, epistemic norms can have effects on our behaviour because we voluntarily follow those norms. But epistemic voluntarism seems to be in conflict with the observation that our epistemic behaviour is normally not subject to our voluntary control.

Descartes has been cited as a typical proponent of epistemic voluntarism by Goldman, Alston and other authors. I argue that Descartes was not an epistemic voluntarist and that his epistemology does not presuppose this brand of voluntarism. In particular, I show that Descartes rejected voluntarism with respect to the two modes of epistemic behaviour most central to Descartes’ epistemology, viz.judgement and suspension of judgement. Finally Descartes’ account of the influence of norms on our epistemic behaviour turns out to be very close to Goldman’s own account.

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Possible Worlds Semantics for Modal Notions Conceived as Predicates (with Hannes Leitgeb and Philip Welch), Journal of Philosophical Logic 32 (2003), pp. 179-223

If L (or the box symbol of modal logic) is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where L is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate L, we tackle both problems. Given a frame (W,R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret L at every world in such a way that L[A] holds at a world w in W if and only if A holds at every world v in W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several paradoxes’ (like Montague’s Theorem, Gödel’s Second Incompleteness Theorem, McGee’s Theorem on the omega-inconsistency of certain truth theories etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of L at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.

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Can We Grasp Consistency?, in The Epistemology of Keith Lehrer, Erik Olsson (ed.), Philosophical Studies Series 95, Kluwer, Dordrecht, 2003, pp. 75-87

In this paper, I will scrutinize a family of arguments which are supposed to show that consistency is not accessible in an epistemically relevant sense. If these arguments were sound, then consistency would be on a par with external facts: neither are directly accessible to us. External factors are not directly accessible because they need to be mediated in some way. Consistency would be inaccessible, according to these arguments, because proving or determining consistency in the relevant cases exceeds our intellectual capabilities. Thus, if consistency were actually inaccessible, then consistency could hardly play a role in an internalist account of epistemology. Traditionally (for instance, in Schlick’s account 1934), however, consistency has been seen as a main ingredient of coherence. Modern epistemologists, like BonJour 1985, have also used consistency as a criterion of coherence, and they have even used consistency in order to define coherence.

The arguments against the accessibility of consistency jeopardizes this central role of consistency in any internalist account of epistemic justification. Thus, epistemic justification cannot imply then consistency of the belief system. If consistency is not directly accessible, then it cannot be used as a criterion for consistency on an internalist account.

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Bealers Masterargument: ein Lehrstück zum Verhältnis von Metaphysik und Semantik (with Holger Sturm), Facta Philosophica 6 (2004), pp. 97-110

In his article Universals (Journal of Philosophy 90, 1993, 5-32) George Bealer puts forward an argument that is supposed to establish two claims: first, universals like propositions, properties and relations do exist, and second universals have to be understood in a Platonic way, that is, universals exist ante rem independently from singular objects. Bealer’s argument, which we call the Master Argument, is based on certain premises. In particular, “that”-sentences are conceived as singular terms, which are obtained from the sentences by (intensional) abstraction. The Master Argument poses a problem for the analysis of so called transmodal sentences. In such sentences intensional abstraction terms occur in the scope of necessity operators. Moreover, such sentences involve de re-modalities. According to the Master Argument a satisfactory semantic analysis of these transmodal sentences requires ante rem universals.

We agree with Bealer that the Master Argument applies against certain kinds of in rebus realism (i.e., Aristotelean conceptions of universals) and against certain kinds of type nominalism, which seeks to replace universals by types of expression. The Master Argument, however, does not refute all variants of type nominalism and in rebus realism. We sketch a version of type nominalism that escapes Bealer’s Master Argument.

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Computational Structuralism (with Leon Horsten), Philosophia Mathematica 13 (2005) pp.174-186

According to structuralism in philosophy of mathematics, arithmetic is about a certain structure. First-order theories are satisfied by (nonstandard) models that do not exhibit this structure. Proponents of structuralism have put forward various accounts of how we succeed to fix one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum’s theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On our account, the intended models of arithmetic are the notation systems which are computably intertranslatable with our notation system of the Arabic numerals.

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The Deflationist’s Axioms for Truth (with Leon Horsten), in Deflationism and Paradox, JC Beall and Brad Armour-Garb (eds.), Oxford University Press, 2005, 203-217

From a deflationist point of view, axiomatic theories of truth seem more appropriate than semantic approaches to truth. The question then arises which set of axioms for truth the deflationist ought to adopt. While there probably does not exist a single best axiomatic theory of truth, it is argued that the truth theory FS that was proposed by Friedman and Sheard looks more promising than is commonly appreciated.

Axiomatizing Kripke’s Theory of Truth (with Leon Horsten), Journal of Symbolic Logic 71 (2006), 677-712

We investigate axiomatizations of Kripke’s theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman’s axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke’s semantics in the straightforward sense; only the sentence provably true in KF are valid in Kripke’s partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt’s conjecture, and provide a direct axiomatization PKF of Kripke’s theory in partial logic. We argue that any natural axiomatization of Kripke’s theory in Strong Kleene logic has the same proof-theoretic strength as PKF , namely the strength of ramified analysis up to \omega^{\omega} and thus any such axiomatization is much weaker than Feferman’s axiomatization KF in classical logic, which is equivalent to ramified analysis up to \epsilon_0.

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How not to state the T-sentences, Analysis 66 (2006), 276-280

In the context of a theory allowing for diagonalisation, the set of the T-sentences T[A]<->A is inconsistent, because of the Liar paradox. Often it is assumed that consistency can be restored by imposing the condition that the sentence A must not contain the truth predicate T. If the underlying language contains also a predicate of necessity (or another modality) and certain natural assumption are made on necessity, then the restricted T-sentences are shown still to yield an inconsistency, although the theory of truth and the theory of necessity taken separately a re perfectly consistent.

The Analysis typesetter introduced a mistake when converting my file into their format. I corrected the mistake in the proofs but this was not taken into account in the printed version. A correction appeared in vol. 67, 268.

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On a Side Effect of Solving Fitch's Paradox by Typing Knowledge, Analysis 68 (2008), 114-120

It has been proposed to block Fitch's paradox by disallowing a predicate or sentential operator of knowledge that can be applied to sentences containing the same predicate or operator of knowledge. Furthermore it has been claimed that this move is not ad hoc as there is independent motivation for this restriction, because this restriction provides a solution also to paradoxes arising from self-reference like the paradox of the Knower. Such a solution is only needed if knowledge is treated as a predicate that can be diagonalized. However, if knowledge and possibility are conceived as such predicates with type restrictions, a new paradox arises. Very basic, jointly consistent assumptions on the predicates of knowledge and possibility yield an inconsistency if (a typed version of) the verifiability principle is added. Thus Fitch's paradox cannot be resolved by typing knowledge.

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Necessities and Necessary Truths: A Prolegomenon to the Metaphysics of Modality, (with Philip Welch), Mind 118 (2009), 71-100

Necessity is usually conceived as a sentential operator rather than a predicate. An intensional sentential operator does not allow one to express quantified statements such as ‘There are necessary a posteriori propositions’ or ‘All laws of physics are necessary’ in the framework of pure first-order logic. Replacing the operator conception of necessity (and other intensional notions) by the predicate conception causes various problems and forces one to reject many philosophical accounts involving necessity. It is argued that the expressive power of the predicate account might be restored if a truth predicate is added to the language of first-order modal logic; then the predicate ‘is necessary’ can be replaced by ‘is necessarily true’. We prove a result showing that this substitution is technically feasible by providing possible-worlds-semantics for the language with a predicate of necessity and performing the reduction of necessity to necessary truth.

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On the Benefits of a Reduction of Modal Predicates to Modal Operators, in Reduction -- Abstraction -- Analysis, proceedings of the 31st International Wittgenstein Symposium Kirchberg, Ontos Verlag, Heusenstamm bei Frankfurt a.M, 2009, 323-333.

A unary predicate is an expression that can be combined with a singular to give a formula. A unary operator is an expression that yields combined with a formula another formula. I consider whether formalising notions such as necessity, analyticity, and aprioricity should be formalised as unary operators or predicates. On the operator account quantified sentences are not easily formalised. I advocate a reduction of the predicate account to an operator conception of modalities. Benefits include the applicability of possible-worlds semantics to modal operators, avoidance of various paradoxes arising from a predicate conception, and ontological parsimony.

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Reducing Compositional to Disquotational Truth, Review of Symbolic Logic 2 (2009), 786-798

Usually disquotational theories of truth, that is theories of truth based on the T-sentences or similar equivalences as axioms are often thought to be deductively weak. This is correct if the truth predicate is allowed to apply only to sentences not containing the truth predicate. By taking a slightly more liberal approach towards the paradoxes, I obtain a disquotational theory of truth that is proof-theoretically as strong as strong compositional theories such as the Kripke-Feferman theory although it doesn't coincide with these theories.

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The Henkin sentence (with Albert Visser), The Life and Work of Leon Henkin (Essays on His Contributions), María Manzano, Ildiko Sain and Enrique Alonso (eds), Studies in Universal Logic, Birkhäuser, to appear

In this paper we discuss Henkin's question concerning a formula that has been described as expressing its own provability. We analyze enkin's formulation of the question and the early repsonses by Kreisel and Löb and sketch how this discussion led to the development of provability Logic. We argue that, in addition to that, the question has philosophical aspects that are still interesting.

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Norms for reflexive theories of truth (with Leon Horsten), in Unifying the Philosophy of Truth, Dora Achourioti, Henri Galinon, Kentaro Fujimoto & José Martínez-Fernández (eds), Springer, to appear

In the past two decades we have witnessed a shift to axiomatic theories of truth. But in this tradition there has been a proliferation of truth theories. In this article we carry out a meta-theoretical reflection on the conditions that we should want axiomatic truth theories to satisfy.

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Self-reference in Arithmetic I (with Albert Visser), Review of Symbolic Logic, to appear

A Gödel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence of arithmetic to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed points for the formulae are obtained. This paper is the first of two papers. In the present paper we focus on provability. In part II, we will consider other properties like Rosser provability and partial truth predicates.

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Self-reference in Arithmetic II (with Albert Visser), Review of Symbolic Logic, to appear

In this sequel to \emph{Self-reference in arithmetic I} we continue our discussion of the question: What does it mean for a sentence of arithmetic to ascribe to itself a property? We investigate how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the expressing formulae are obtained. In this second part we look at some further examples. In particular, we study sentences apparently expressing their Rosser-provability, their own $\Sigma^0_n$-truth or their own $\Pi^0_n$-truth. Finally we offer an assessment of the results of both papers.

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Axiomatizing Semantic Theories of Truth? (with Martin Fischer, Jönne Speck and Johannes Stern)

We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ω-categoricity and discuss its usefulness and limits.